FIXME: I'd like a better reason/feel for why you can differentiate and
integrate termwise: it's true for a good reason, not just a
calculation.
(it's basically because of the uniform convergence, but that's not
enough (e.g., $\frac{1}{n} \sin (n^2 x)$ converges uniformly to $0$,
but its derivative goes haywire)
